Optimal. Leaf size=209 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (c^2 f^2 \left (m^2+3 m+2\right )-2 c d e f (m+1) (m+n+3)+d^2 e^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b d^2 (m+1) (m+n+2) (m+n+3)}-\frac{f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d^2 (m+n+2) (m+n+3)}+\frac{f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)} \]
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Rubi [A] time = 0.18416, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {90, 80, 66, 64} \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \left (c^2 f^2 \left (m^2+3 m+2\right )-2 c d e f (m+1) (m+n+3)+d^2 e^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )\right ) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b d^2 (m+1) (m+n+2) (m+n+3)}-\frac{f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d^2 (m+n+2) (m+n+3)}+\frac{f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)} \]
Antiderivative was successfully verified.
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Rule 90
Rule 80
Rule 66
Rule 64
Rubi steps
\begin{align*} \int (b x)^m (c+d x)^n (e+f x)^2 \, dx &=\frac{f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac{\int (b x)^m (c+d x)^n (-b e (c f (1+m)-d e (3+m+n))-b f (c f (2+m)-d e (4+m+n)) x) \, dx}{b d (3+m+n)}\\ &=-\frac{f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac{f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac{\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) \int (b x)^m (c+d x)^n \, dx}{d^2 (2+m+n) (3+m+n)}\\ &=-\frac{f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac{f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac{\left (\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) (c+d x)^n \left (1+\frac{d x}{c}\right )^{-n}\right ) \int (b x)^m \left (1+\frac{d x}{c}\right )^n \, dx}{d^2 (2+m+n) (3+m+n)}\\ &=-\frac{f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac{f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac{\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) (b x)^{1+m} (c+d x)^n \left (1+\frac{d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac{d x}{c}\right )}{b d^2 (1+m) (2+m+n) (3+m+n)}\\ \end{align*}
Mathematica [A] time = 0.231982, size = 153, normalized size = 0.73 \[ \frac{x (b x)^m (c+d x)^n \left (f (c+d x) (e+f x)-\frac{\left (\frac{d x}{c}+1\right )^{-n} (d e (m+n+2) (c f (m+1)-d e (m+n+3))-c f (m+1) (c f (m+2)-d e (m+n+4))) \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )+f (m+1) (c+d x) (c f (m+2)-d e (m+n+4))}{d (m+1) (m+n+2)}\right )}{d (m+n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2} \left (b x\right )^{m}{\left (d x + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} \left (b x\right )^{m}{\left (d x + c\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 16.892, size = 131, normalized size = 0.63 \begin{align*} \frac{b^{m} c^{n} e^{2} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac{2 b^{m} c^{n} e f x^{2} x^{m} \Gamma \left (m + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} + \frac{b^{m} c^{n} f^{2} x^{3} x^{m} \Gamma \left (m + 3\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 3 \\ m + 4 \end{matrix}\middle |{\frac{d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 4\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2} \left (b x\right )^{m}{\left (d x + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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